The polynomials f(x) = ax³ + 3x² - 3 and g(x) = 2x³ - 5x + a when divided by (x - 4) leave the same remainder in each case. Find the value of a.

By remainder theorem,

If f(x) is divided by (x - a), then remainder = f(a).

Given,

f(x) = ax³ + 3x² - 3

g(x) = 2x³ - 5x + a

Divisor :

⇒ x - 4 = 0

⇒ x = 4

On dividing ax³ + 3x² - 3 by x - 4,

⇒ f(4) = a(4)³ + 3(4)² - 3

= 64a + 48 - 3

= 64a + 45.

On dividing 2x³ - 5x + a by x - 4,

⇒ g(4) = 2(4)³ - 5(4) + a

= 128 - 20 + a

= 108 + a.

Given,

On dividing by (x - 4) polynomials f(x) = ax³ + 3x² - 3 and g(x) = 2x³ - 5x + a leave same remainder.

⇒ f(4) = g(4)

⇒ 64a + 45 = 108 + a

⇒ 64a - a = 108 - 45

⇒ 63a = 63

⇒ a = $\dfrac{63}{63}$

⇒ a = 1.

Hence, the value of a = 1.